Exponential functions and logarithms: Logarithmic functions
Transformations of the logarithmic function
Previously, we looked at the graph of the logarithmic function. We can also transform the graphs, just like we can with the exponential function. For every transformation, we will give the corresponding asymptote, the domain and the range. Keep in mind that the asymptote of the normal logarithmic function is the #y#axis, in other words the line #x=0#. The domain is all numbers greater than #0# and the range is all numbers.
Transformations
We can transform the formula #y=\log_{\blue{a}}\left(x\right)# in three ways. In all three of the images, the dotted line is the graph of the function #\log_{\blue{a}}\left(x\right)#.
Transformation  Examples  
1 
We shift the graph of #y=\log_{\blue{a}}\left(x\right)# up with #\green{b}#. The new formula becomes \[\log_{\blue{a}}\left(x\right)+\green b\] The asymptote, the domain and the range all stay the same. 
geogebra

2 
We shift the graph of #\log_{\blue{a}}\left(x\right)# to the right with #\purple{c}#. The new formula becomes \[\log_{\blue{a}}\left(x\purple{c}\right)\] The asymptote is now #x=\purple{c}#, the domain is all numbers greater than #\purple{c}# and the range is all numbers. 
geogebra

3 
We multiply the graph of #y=\log_{\blue{a}}\left(x\right)# with #\orange{d}# relative to the #x#axis. The new formula becomes \[y=\orange{d}\cdot\log_{\blue{a}}\left(x\right)\] The asymptote, the domain and the range all stay the same. 
geogebra

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